We study p-Faulty Search, a variant of the classic cow-path optimization problem, where a unit speed robot searches the half-line (or 1-ray) for a hidden item. The searcher is probabilistically faulty, and detection of the item with each visitation is an independent Bernoulli trial whose probability of success p is known. The objective is to minimize the worst case expected detection time, relative to the distance of the hidden item to the origin. A variation of the same problem was first proposed by Gal [28] in 1980. Alpern and Gal [3] proposed a so-called monotone solution for searching the line (2-rays); that is, a trajectory in which the newly searched space increases monotonically in each ray and in each iteration. Moreover, they conjectured that an optimal trajectory for the 2-rays problem must be monotone. We disprove this conjecture when the search domain is the half-line (1-ray). We provide a lower bound for all monotone algorithms, which we also match with an upper bound. Our main contribution is the design and analysis of a sequence of refined search strategies, outside the family of monotone algorithms, which we call t-sub-monotone algorithms. Such algorithms induce performance that is strictly decreasing with t, and for all p ∈ (0, 1). The value of t quantifies, in a certain sense, how much our algorithms deviate from being monotone, demonstrating that monotone algorithms are sub-optimal when searching the half-line.
We study a primitive vehicle routing-type problem in which a fleet of n unit speed robots start from a point within a non-obtuse triangle ∆, where n ∈ {1, 2, 3}. The goal is to design robots' trajectories so as to visit all edges of the triangle with the smallest visitation time makespan. We begin our study by introducing a framework for subdividing ∆ into regions with respect to the type of optimal trajectory that each point P admits, pertaining to the order that edges are visited and to how the cost of the minimum makespan R n (P ) is determined, for n ∈ {1, 2, 3}. These subdivisions are the starting points for our main result, which is to study makespan trade-offs with respect to the size of the fleet. In particular, we define R n,m (∆) = max P ∈∆ R n (P )/R m (P ), and we prove that, over all non-obtuse triangles ∆: (i) R 1,3 (∆) ranges from 10 to 4, (ii) R 2,3 (∆) ranges from 2 to 2, and (iii) R 1,2 (∆) ranges from 5/2 to 3. In every case, we pinpoint the starting points within every triangle ∆ that maximize R n,m (∆), as well as we identify the triangles that determine all inf ∆ R n,m (∆) and sup ∆ R n,m (∆) over the set of non-obtuse triangles.
Background: Vineland adaptive behavior scales-parent/caregiver rating form (VABS II) is a questionnaire used to examine adaptive behavior in individuals, whose age ranges from birth to 90 years old. The purpose of this study was to translate and assess the psychometric properties of the Greek version VABS II-parents/caregiver, in children. Method: Three samples of participants within the ages 5 -10 years were analyzed; including two groups of developmental disorders (N = 116) and control group (N = 90). The questionnaire was translated into Greek by two bilingual translators. The pre-final version was pilot tested in 30 mothers of typical and atypical development children, aged 3 -10 years. The final version was submitted in 206 subjects, twice, in different ways for reliability testing. A split-half reliability test was employed for the reliability of scores for two halves of the test, to evaluate the reliability and internal consistency, of the VABS II-Gr. The spearman-brown formula was used to determine correlations between the domains. Α knowngroup method was utilized, to estimate construct validity, exploring the differences between the two groups. Results: Across the age groups, overall, the domain reliability estimates are quite high, with a value of .83 to .95. Equivalence reliability (correlation) was found to be excellent (r = .90). Conclusion:
The input to the Triangle Evacuation problem is a triangle ABC. Given a starting point S on the perimeter of the triangle, a feasible solution to the problem consists of two unit-speed trajectories of mobile agents that eventually visit every point on the perimeter of ABC. The cost of a feasible solution (evacuation cost) is defined as the supremum over all points T of the time it takes that T is visited for the first time by an agent plus the distance of T to the other agent at that time. Similar evacuation type problems are well studied in the literature covering the unit circle, the p unit circle for p ≥ 1, the square, and the equilateral triangle. We extend this line of research to arbitrary non-obtuse triangles. Motivated by the lack of symmetry of our search domain, we introduce 4 different algorithmic problems arising by letting the starting edge and/or the starting point S on that edge to be chosen either by the algorithm or the adversary. To that end, we provide a tight analysis for the algorithm that has been proved to be optimal for the previously studied search domains, as well as we provide lower bounds for each of the problems. Both our upper and lower bounds match and extend naturally the previously known results that were established only for equilateral triangles.
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